L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
1
MER301: Engineering Reliability
LECTURE 16:
Measurement System Analysis and Uncertainty Analysis-Part 1
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
We must submit the output from our design process to a second (measurement) process
MeasurementProcess
Outputs• Measurements
ProcessInputs Outputs
Inp
uts
Parts(Example)
Measurement as a Process
2
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MER301: Engineering ReliabilityLecture 16
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Measurement System Concerns..
How big is the measurement error? What are the sources of measurement error? Are the measurements being made with units
which are small enough to properly reflect the variation present?
Is the measurement system stable over time? How much uncertainty should be attached to a
measurement system when interpreting data from it?
How do we improve the measurement system?
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Measurement System Analysis Total Error in a measurement is defined as the difference
between the Actual Value and Observed Value of Y Two general categories of error – Accuracy or Bias and
Precision Accuracy or Bias of Measurement System is defined as the difference
between a Standard Reference and the Average Observed Measurement Precision of a Measurement System is defined as the standard deviation of
Observed Measurements of a Standard Reference Total Error = Bias Error + Precision Error for independent random variables
Measurement System Error is described by Average Bias Error (Mean Shift)and a statistical estimate of the Precision Error (Variance)
Measurement System Analysis is a Fundamental Part of Every Experiment
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MER301: Engineering ReliabilityLecture 16
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Not Accurate, Not Precise Accurate, Not Precise
Not Accurate, Precise Accurate, Precise
Precision and Accuracy
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MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Bias or Accuracy error is a constant value and is dealt with by calibrating the measurement system
Variation or Precision error is a random variable which depends on the measurement equipment(the instruments used) and on the measurement system repeatability and reproducibility. Instrument Capability Analysis, Test/retest (repeatability)and Gage R&R studies are used to quantify the size of these errors.
222 0
0
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved YYY
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.
= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
7
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Example 16.1-Effect of Measurement System Variation
Calculate the effect of measurement system variation on the acceptance rate for a part with USL and LSL at Z= +/-1.96 respectively. If then what is the percentage of acceptable parts that will be rejected? If on the other hand what is the percentage of acceptable parts that will be rejected?
2/221 actm
20/222 actm
LSL USL
T
Process
Gauge
Failed Goodunits
Gauge
Passed BadUnits
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
9
Impact of Measurement System Variation on Variation in Experimental Data…
LSL USL
T
Process
Gauge
Failed Goodunits
Gauge
Passed BadUnits
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Example 16.1(con’t: ) For the product, the spec limits of +/-1.96 mean that the
2.5% of parts in each tail are out of spec. Thus
For the observed standard deviation is
and
Then the acceptable parts now rejected are
act
actuslusl
XZ
96.1
2/221 actm
2/32/22221 actactactmactobs
60.12/3
96.1
2/3
)96.1(,
act
actactactObserveduslZ
060.0945.0975.0260.196.12
actactuslX 96.1
2/221 actm
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Example 16.1(con’t: ) For the product, the spec limits of +/-1.96 mean that the
2.5% of parts in each tail are out of spec. Thus
For the observed standard deviation is
and
Then the acceptable parts now rejected are
act
actuslusl
XZ
96.1
20/222 actm
actactactactobs 0247.120/1120/22
913.10247.1
96.1
0247.1
)96.1(,
act
actactactObserveduslZ
006.0972.0975.02913.196.12
actactuslX 96.1
20/222 actm
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Example 16.1(con’t)
Process2
2 2Measure
Process2
2 20Measure
Measure
Measure
Process
Process
Observed
Observed
Set Measurement System Requirements Based on the Process Variation
Unacceptable
Acceptable
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
Gage Repeatability & Reproducibility ---GRR or GR&R---
Gage Repeatability & Reproducibility compares measurement system variation and product variation
The term is the size of an interval containing 99% of the measured values made on a specific item
The Tolerance- often equal to - is the size of the interval where a product has acceptable dimensions, performance, or other characteristics
%100% 15.5 Tolerance
measurmentGRR
tmeasuremen15.5
actual6
13
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MER301: Engineering ReliabilityLecture 16
Gage Performance relative to required Tolerance Band
R&R less than 10% - Measurement system is acceptable. R&R 10% to 30% - Maybe acceptable - make decision
based on classification of characteristic, hardware application, customer input, etc.
R&R over 30% - Not typically acceptable. Find the problem using root cause analysis(fishbone), remove root causes
GRR is a measure of “noise” in the data GRR is a measure of “noise” in the data
%100% 15.5 Tolerance
measurmentGRR
14
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Effect of Gage R&R on Variation
GRR <10% means < 0.7% of the variation in the experimental data is from the measurement system
GRR> 30% means that > 5.9% of the variation in the experimental data is from the measurement system
%1006
15.5%
actual
tmeasuremenGRR
2
222
)15.5/6(1/ GRRactualobserved
tmeasuremenactualobserved
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MER301: Engineering ReliabilityLecture 16
16
GRR Example 16.2
607.06
2/15.5
6
2/15.5
6
15.51
1
act
act
act
mGRR
192.06
20/15.5
6
20/15.5
6
15.52
2
act
act
act
mGRR
The GRR values for the previous Example 16.1 are
The capabilities of two (or more) measurement systems can be compared by comparing the GRR’s for each. Since GRR2<GRR1 , the second measurement system is more capable than the first. The observed standard deviations quantify how much better….
20/222 actm
2/221 actm
195.1/21obsobs 0247.1/
225.1/
2
1
actobs
actobs
8365.0/12obsobs
L Berkley DavisCopyright 2009
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GRR Example 16.2
607.06
2/15.5
6
2/15.5
6
15.51
1
act
act
act
mGRR
192.06
20/15.5
6
20/15.5
6
15.52
2
act
act
act
mGRR
The GRR values for the previous Example 16.1 are at best marginally acceptable(GRR2 ) or not acceptable(GRR1 )
For a GRR value equal to 10% (0.10) there results
20/222 actm
2/221 actm
74/167.73/1)15.5/610.0(/6
15.510.0 222
actmact
mGRR
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MER301: Engineering ReliabilityLecture 16
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Example 16.2 ( so ) For the product, the spec limits of +/-1.96 mean that the
2.5% of parts in each tail are out of spec. Thus
For the observed standard deviation is
and
Then the acceptable parts now rejected are
act
actuslusl
XZ
96.1
74/22actm
actactactactobs 0067.174/1174/22
947.10067.1
96.1
0067.1
)96.1(,
act
actactactObserveduslZ
0016.09742.09750.02947.1960.12
actactuslX 96.1
74/22actm 10.0GRR
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Summarizing how it all fits together…..
When a set of measurements are made, the results are always observed values,
If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated
If the item being measured is a standard reference
If the measurement system bias and variance are known then the actual mean and actual variance can be calculated
mbiasactobs YYY 222 0 mactobs
actobsbias 222actobsm
222mobsact
022 obsm
0 biasactobs
biasobsact
%100% 15.5 Tolerance
measurmentGRR
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
Total Variation made up of Actual Process Variation and Measurement
System Variation
Total Variation made up of Actual Process Variation and Measurement
System Variation
Sources of Measurement System Error
ProcessInputs Outputs Inputs MeasurementProcess
Outputs
• Observations• Measurements
Long-term Process Variation
Actual Process Variation
Accuracy (Bias)
Accuracy (Bias)
Measurement VariationMeasurement Variation
Observed Process Variation
Short-term Process Variation
Variation due to gauge
Variation due to gauge
Variation due to operator
Variation due to operator
Precision (Pure Error)
Precision (Pure Error)
Stability (time dependent)
Stability (time dependent)
Linearity (value dependent)
Linearity (value dependent)
RepeatabilityRepeatability
ReproducibilityReproducibility
within sample variation
within sample variation
MeasurementSystem
Repeatability
Resolution
20
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Measurement System Errors
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average
(Low End)
Observed Average
(High End)
Linearity
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Elements that contribute to Accuracy and Precision Errors
Instrument Capability Resolution Gage Repeatability Linearity
Measurement System - Short Term (ST) Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)
Measurement System - Long Term (LT) Use Measurement System - Short Term Use Reproducibility Stability
First Two are Entitlement….Third is Reality
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Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Elements that contribute to Precision or Variation Errors
Instrument Capability Resolution Gage Repeatability Linearity
Measurement System- Short Term (ST) Use Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)
Measurement System - Long Term (LT) Use Measurement System - Short Term Use (ST) Reproducibility(Gage R&R) Stability(Gage R&R)
First Two are Entitlement….Third is Reality
2instrument
222, ityrepeatibilinstrumentSTtmeasuremen
22222ilityreproducibityrepeatabilinstrumentLTm
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
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MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
2222
222
0
0
0
ilityreproducibityrepeatabilinstrumenttmeasuremen
Y
ilityreproducibityrepeatabilinstrumenttmeasuremen
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved
tmeasuremen
YYYY
YYY
From pages119-120…
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Updating how variances all fit together
When a set of measurements are made, the results are always observed values,
If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated
If the item being measured is a standard reference
If the measurement system bias and variance are known then the actual mean and actual variance can be calculated
mbiasactobs YYY 2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs
222mobsact
22222 0 ilityreproducibityrepeatabilinstrumentobsm
222222ilityreproducibityrepeatabilinstrumentactobsm
)( 22222ilityreproducibityrepeatabilinstrumentobsact
biasactualobserved
biasobsact
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MER301: Engineering ReliabilityLecture 16
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Emissions Sampling
NOxInstrument Yactual-
NOx fromGas turbine
Cal/ZeroGases
Yobs- NOx Reading
Heated Sampling Line
Calibration Gas
Sample Conditioning
tmeasuremenbiasactualobserved YYY
222tmeasuremenactobs
biasactobs
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Elements that contribute to Accuracy and Precision Errors
Instrument CapabilityResolutionGage RepeatabilityLinearityMeasurement System- Short Term(ST) UseInstrument CapabilityEquipment CalibrationTest/Re-Test StudyMeasurement System- Long term (LT) UseMeasurement System -Short Term(ST) Use ReproducibilityStability 2222
ilityreproducibityrepeatabilinstrumenttmeasuremen
Union CollegeMechanical Engineering
MER301: Engineering ReliabilityLecture 16
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Emissions Sampling
NOxInstrument Yactual-
NOx fromGas turbine
Cal/ZeroGases
Yobs- NOx Reading
Heated Sampling L ine
Calibration Gas
Sample Conditioning
tmeasuremenbiasactualobserved YYY
222tmeasuremenactobs
biasactobs
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
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Establish magnitude and sources of
measurement system error due to bias and precision errors
Tools Instrument Capability Analysis Test/Re-test – system precision/repeatability Calibration - bias “Continuous Variable” Gage R&R (Gage
Reproducibility and Repeatability) Attribute Variable Gage R&R Destructive Gage R&R
How Can we Address Accuracy and Precision Errors?
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
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Measurement System Analysis
Instrument Capability Analysis….. Resolution-smallest increment that the gage can resolve in the
measurement process. Gage should be able to resolve tolerance band into ten or more parts. Resolution Uncertainty =
Instrument Accuracy- measure of instrument repeatability or instrument “noise”.. Found by repeated measurements of the same test item. Uncertainty =
Linearity- consistency of the measurement system across the entire range of the measurement system. Linearity Uncertainty =
The variations are combined as follows
00 4 u
rru 4
llu 4
2222
2222
lroinstrument
lroinstrument uuuu
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
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Instrument Capability Analysis….. Variation for any one instrument equals the sum of the
resolution, repeatability and linearity terms
The Variation for “n” instruments equals the sum of the variations for each individual instrument
Each of the “n” instruments has resolution, repeatability, and linearity terms that must be taken into account
2222linearityityrepeatabilresolutioninstrument
linearityityrepeatabilresolutioninstrument YYYY
222221
21
n
n
instrumentinstrumentinstrumentinstrument
instrumentinstrumentinstrumentinstrument YYYY
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Instrument Capability Analysis - Resolution of Instruments/Sensors
The measurement uncertainty due to resolution is generally taken as a specified fraction of the smallest increment an instrument can resolve, ie as a fraction of the smallest scale division
General Rule: assign a numerical value for the mean value of equal to one half of the instrument resolution. This means
that half of the smallest scale division is assumed to equal a 95% Confidence Interval ( a wide band) for variation due to resolution
resolution 4 21
0 ou
o4
0u
0u
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Instrument Capability Analysis Repeatability and Linearity
The manufacturer of an instrument will provide information on the capability of the instrument in the specification sheets provided with the instrument
The numerical values given for Instrument or Sensor Accuracy and Linearity are almost always uncertainties Let = uncertainty due to the equipment
accuracy/repeatability error where Let = uncertainty due to linearity error
where The inherent capability/uncertainty of the instrument/sensor
is then estimated as:
222
lroinstrument uuuu
rurru 4
lullu 4
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Example 16.3-Instrument Capability The Capability of a force measuring instrument is
described by catalogue data. Calculate an estimate of the variation attributable to this instrument. Express the result both in dimensional terms (N) and in dimensionless terms for a reading R=50N
Resolution 0.25NRange 0 to 100NLinearity within 0.20N over rangeRepeatability within 0.30N over range
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Example 16.3(con’t) An estimate of the instrument uncertainty depends
on the combined uncertainties due to resolution, repeatability and linearity
The instrument uncertainty is then
0.0076 50
38.0
0.38N)125.0()3.0()2.0( 222
N
N
R
uu
u
dinstrument
instrument
Nu rr 30.04
N 125.02/25.04 oou
Nu ll 20.04
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Instrument Capability Analysis Summary….. Resolution-smallest increment that the gage can resolve in the
measurement process. Gage should be able to resolve tolerance band into ten or more parts. Resolution Uncertainty =
Instrument Accuracy- measure of gage repeatability or gage “noise”.. Found by repeated measurements of the same test item. Uncertainty =
Linearity- consistency of the measurement system across the entire range of the measurement system. Linearity Uncertainty =
The variations are combined as follows
00 4 u
rru 4
llu 4
2222
2222
lroinstrument
lroinstrument
uuuu
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Measurement System Short Term Use Includes Instrument Capability Repeatability - variation when one operator repeatedly
makes the same measurement with the same measuring equipment Test/Re-test Study
Calibration/Bias
Measurement System-Long Term Use Includes Measurement System –Short Term Use Reproducibility- variation when two or more operators make
same measurement with the same measuring equipment Stability-variation when the same operator makes the same
measurement with the same equipment over an extended period of time
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Test/Retest Example 16.4 Test/Retest (Repeatability) Study on a
Measurement System. Thirty repeat measurements were taken on a Standard Reference Item with a thickness of 50mils The tolerance band for the application is 20mils(+/-10).
Data, in mils 53,45,52,47,54,52,52,55,52,48,48,53,55,51,47,52,47,35,
45,54,48,51,53,44,52,52,55,59,53,53
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Example 16.4(con’t) Objective is to establish the precision and accuracy
of the measurement system Precision-Repeatability
In a good Measurement System, 99% of the measurements of a given item should fall within a band less than
1/10 of tolerance band
Accuracy/Bias Bias = sample mean- true value
LSL USL
T
Process
Gauge
Failed Goodunits
Gauge
Passed BadUnits
10/115.5
tolerance
GRR tmeasuremen
actualobserved YYbias
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MER301: Engineering ReliabilityLecture 16
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Example 16.4 Run Chart and Histogram
10 20 30
40
50
60
Observation
C1
Number of runs about median:Expected number of runs:Longest run about median:Approx P-Value for Clustering:Approx P-Value for Mixtures:
Number of runs up or dow n:Expected number of runs:Longest run up or dow n:Approx P-Value for Trends:Approx P-Value for Oscillation:
13.000014.9333 6.0000 0.2190 0.7810
19.000019.6667 3.0000 0.3829 0.6171
Run Chart for C1
35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0
0
5
10
C1
Fre
que
ncy
These results look bad to the eye…there are outliers and mean is high
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Test/Retest Study Example 16.4 Summary
Descriptive Statistics
Variable N Mean Median StDev SE Mean
C1 30 50.567 52.000 4.561 0.833
Variable Minimum Maximum Q1 Q3
C1 35.000 59.000 47.750 53.000
Conclusions Given the tolerance band of 20 mils,there is an
unacceptable level of device precision
Given the Reference Test item had a known thickness of 50mils, the bias(inaccuracy) is:
bias = 50.57 – 50.0 = 0.57mils
)10/1(174.120/48.2320/15.5 tmeasuremenGRR
bias
bias
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MER301: Engineering ReliabilityLecture 16
45
Not Accurate, Not Precise Accurate, Not Precise
Not Accurate, Precise Accurate, Precise
Example 16.3- Precision versus Accuracy
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Measurement System Analysis
Measurement System-Short Term Use Repeatability-variation when one operator repeatedly
makes the same measurement with the same measuring equipment Test/Re-test Study
Measurement System - Long Term Use Reproducibility- variation when two or more operators
make same measurement with the same measuring equipment
Stability-variation when the same operator makes the same measurement with the same equipment over an extended period of time
L Berkley DavisCopyright 2009
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Elements that contribute to Accuracy and Precision Errors
Instrument Capability Resolution Gage Repeatability Linearity
Measurement System - Short Term (ST) Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)
Measurement System - Long Term (LT) Use Measurement System - Short Term Use Reproducibility Stability
First Two are Entitlement….Third is Reality
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
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Measurement System Analysis: A Summary of the Basic Equations
22222
2222
222
2222
222 0
0
ilityreproducibityrepeatabilinstrumentactualobserved
ilityreproducibSTtmeasuremenLT
ityrepeatabilinstrumentST
ilityreproducibityrepeatabilinstrumenttmeasuremen
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved YYY
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